A001615 -id:A001615 - cp's OEIS frontend

A351455 a(n) = A064989(A001615(A003961(n))).

1, 1, 2, 2, 1, 2, 2, 4, 6, 1, 5, 4, 4, 2, 2, 8, 3, 6, 2, 2, 4, 5, 6, 8, 5, 4, 18, 4, 1, 2, 17, 16, 10, 3, 2, 12, 10, 2, 8, 4, 7, 4, 2, 10, 6, 6, 8, 16, 14, 5, 6, 8, 6, 18, 5, 8, 4, 1, 29, 4, 13, 17, 12, 32, 4, 10, 4, 6, 12, 2, 31, 24, 3, 10, 10, 4, 10, 8, 10, 8, 54, 7, 12, 8, 3, 2, 2, 20, 25, 6, 8

Offset: 1

Antti Karttunen, Feb 12 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001615, A003557, A003961, A064989, A151800, A347385, A351450.
Coincides with A326042 on squarefree numbers (A005117, and apparently on no other numbers).
Cf. also A351441.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A351455(n) = A064989(A001615(A003961(n)));

Multiplicative with a(p^e) = A064989((q+1)*q^(e-1)), where q = nextPrime(p) = A151800(p).
a(n) = A064989(A001615(A003961(n))) = A064989(A347385(A003961(n))).
a(n) = A003557(n) * A351450(n).

A253628 Psi(n) mod n, where Psi is the Dedekind psi function (A001615).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 4, 3, 8, 1, 0, 1, 10, 9, 8, 1, 0, 1, 16, 11, 14, 1, 0, 5, 16, 9, 20, 1, 12, 1, 16, 15, 20, 13, 0, 1, 22, 17, 32, 1, 12, 1, 28, 27, 26, 1, 0, 7, 40, 21, 32, 1, 0, 17, 40, 23, 32, 1, 24, 1, 34, 33, 32, 19, 12, 1, 40, 27, 4, 1, 0, 1, 40, 45

Offset: 1

Tom Edgar, Jan 06 2015

nonn

a(n) = A054024(n) when n is squarefree.
Indices of 1 appear to be given by primes A000040 (see conjecture in A068494). The (weaker) statement that a(prime(i)) = 1 is a direct consequence of the multiplicity of A001615.
a(n) = 0 if n is a member of A187778.

			A001615(12) = 24 and 24 == 0 (mod 12) so a(12) = 0.
A001615(15) = 24 and 24 == 9 (mod 15) so a(15) = 9.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001615, A068494, A054024, A006093, A187778.

A253628 := proc(n)
    modp(A001615(n),n) ;
end proc: # R. J. Mathar, Jan 09 2015

a253628[n_] :=
Mod[DirichletConvolve[j, MoebiusMu[j]^2, j, #], #] & /@ Range@n; a253628[75] (* Michael De Vlieger, Jan 07 2015, after Jan Mangaldan at A001615 *)

[(n*mul(1+1/p for p in prime_divisors(n)))%n for n in [1..100]]

a(n) = A001615(n) mod n.

A291051 a(n) is the smallest number k such that psi(k) = n*phi(k) where psi(k) is Dedekind psi function (A001615) and phi(k) is Euler totient function (A000010), or 0 if no such k exists.

Original entry on oeis.org

1, 3, 2, 14, 190, 6, 78, 42, 30, 570, 16770, 210, 1102290, 2730, 67830, 43890, 133707210, 746130, 27606810, 16546530, 9699690, 417086670, 3828438543930, 8720021310, 705196562070

Offset: 1

Altug Alkan, Aug 17 2017

nonn

Also a(n) is the smallest squarefree number k such that sigma(k) = n*phi(k), or 0 if no such k exists.
It is conjectured that A055234(n) > 0 for each n. Is a(n) > 0 for all values of n?
10^12 < a(26) <= 50353622409090. - Giovanni Resta, Aug 18 2017

			a(4) = 14 since psi(14) / phi(14) = 24 / 6 = 4 and 14 is the least number with this property.

Cf. A000010, A001615, A055234.

psi[n_] := n*Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; f[n_] := Block[{k = 1}, While[ n*EulerPhi[k] != psi[k], k++]; k]; Array[f, 22] (* Robert G. Wilson v, Sep 15 2017 *)

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
a(n) = {my(k = 1); while (n*eulerphi(k) != a001615(k), k++); k; } \\ Altug Alkan, Aug 17 2017, after Charles R Greathouse IV at A001615

a(23)-a(25) from Giovanni Resta, Aug 18 2017

A291138 a(n) is the smallest k such that psi(k) and phi(k) have same distinct prime factors when k is the product of n distinct primes (psi(k) = A001615(k) and phi(k) = A000010(k)), or 0 if no such k exists.

Original entry on oeis.org

3, 14, 42, 210, 3570, 43890, 746130, 14804790, 281291010, 8720021310, 278196808890, 8624101075590, 353588144099190, 25104758231042490, 2234323482562781610, 129325924468711040070, 9182140637278483844970, 725389110345000223752630, 51501592227099266198116170

Offset: 1

Altug Alkan, Aug 18 2017

nonn

			a(5) = 3570 = 2*3*5*7*17 because psi(3570) = 3*4*6*8*18 = 2^7*3^4, and phi(3570) = 2*4*6*16 = 2^8*3^1 and 3570 is the least number with 5 distinct prime factors having this property.

Daniel Suteu, Table of n, a(n) for n = 1..36

Cf. A000010 (phi), A005117 (squarefree), A001615 (psi), A007947 (radical).

Rest@ Values[#][[All, 1]] &@ KeySort@ PositionIndex@ Table[If[SameQ @@ #, PrimeNu@ n, 0] &@ Map[FactorInteger[#][[All, 1]] &, {EulerPhi@ n, n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]}], {n, 10^6}] (* Michael De Vlieger, Aug 26 2017, after Michael Somos at A001615 *)

generate(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j, phi=1, psi=1) = my(list=List()); my(s=sqrtnint(B\m, j)); if(j==1, forprime(q=max(p, ceil(A/m)), s, if(factorback(factor((q-1)*phi)[, 1]) == factorback(factor((q+1)*psi)[, 1]), listput(list, m*q))), forprime(q=p, s, my(t=m*q); list=concat(list, f(t, q+1, j-1, phi*(q-1), psi*(q+1))))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jan 25 2023

a(10) from Giovanni Resta, Aug 26 2017

a(11)-a(19) from Daniel Suteu, Jan 25 2023

A291205 Numbers n such that psi(n) = phi(m) has no solution for any m (psi(n) = A001615(n) and phi(m) = A000010(m)).

Original entry on oeis.org

2, 13, 37, 50, 58, 61, 67, 73, 74, 89, 97, 111, 113, 122, 151, 157, 169, 173, 181, 183, 193, 229, 233, 241, 250, 257, 259, 274, 277, 283, 298, 307, 313, 314, 317, 337, 349, 353, 373, 386, 389, 394, 397, 401, 409, 421, 427, 433, 449, 453, 457, 466, 481, 487, 507, 509, 514, 541, 543, 547, 554, 557, 562

Offset: 1

Altug Alkan, Aug 21 2017

Numbers n such that psi(n) is not a totient number (A002202) where psi(n) = A001615(n).

			58 is a term because psi(58) = 90 is not a term of A002202.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A001615, A002202.

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
isok(n) = !istotient(a001615(n)); \\ after Charles R Greathouse IV at A001615

A291209 Numbers n such that psi(n) is the sum of proper divisors of n where psi(n) = A001615(n).

Original entry on oeis.org

9504, 16560, 41400, 5048568, 10889856, 11941344, 16255080, 131473152, 5517818880, 107561120688, 612014161920

Offset: 1

Altug Alkan, Aug 21 2017

Or numbers n such that sigma(n) = psi(n) + n where psi(n) = A001615(n) and sigma(n) = A000203(n).

9504 = 2^5*3^3*11 is the smallest number with this property.

			16560 is a term because sigma(16560) = 58032 and psi(16560) = 41472; sigma(16560) - psi(16560) = 16560.

Cf. A000203, A001065, A001615.

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
isok(n) = sigma(n)==a001615(n)+n; \\ after Charles R Greathouse IV at A001615

a(9)-a(11) from Giovanni Resta, Aug 21 2017

A291487 a(n) is the smallest k such that psi(k) = n!, or 0 if no such k exists (psi(k) = A001615(k)).

Original entry on oeis.org

1, 1, 0, 4, 12, 75, 300, 1950, 13650, 122850, 1160250, 13340250, 140390250, 1825073250, 25318743450, 370489869750, 5503458610650, 93558796381050, 1643961707838450, 30815473745606550, 596477734382780250, 12526032422038385250, 272871020017346000250, 6276033460398958005750

Offset: 0

Altug Alkan, Aug 24 2017

nonn

			a(5) = 75 because psi(75) = 120 = 5! and 75 is the least number with this property.
a(7) = 1950 and 1950 has no prime factor 7, so a(8) = 7*1950 = 13650.

Cf. A000142, A001615.

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))
a(n) = my(N=n!); for(k=1, N, if(a001615(k)==N, return(k))); 0 \\ after Charles R Greathouse IV at A001615

a(13)-a(14) from Giovanni Resta, Aug 25 2017

a(15)-a(23) from Daniel Suteu, Dec 29 2020

A291959 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and psi(n) are all integers, where phi(n) is the Euler totient function (A000010) and psi(n) is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 150, 300, 418, 450, 525, 600, 750, 836, 900, 1200, 1350, 1500, 1575, 1672, 1800, 2250, 2400, 2625, 2700, 3000, 3135, 3344, 3600, 3675, 3750, 4050, 4500, 4598, 4725, 4800, 5400, 6000, 6688, 6750, 7200, 7500, 7875, 7942, 8100, 9000, 9196, 9405, 9600, 10800

Offset: 1

Amiram Eldar, Sep 06 2017

nonn

The number of terms below 10^3, 10^4, ... are 10, 44, 147, 397, ...

			phi(150)=40, psi(150)=360, their arithmetic mean = 200, geometric mean = 120, harmonic mean = 72 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A001615, A065146.

dedekindPsi[n_] := If[n < 1, 0, n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]];
aQ[n_] := Module[{s = dedekindPsi[n], p = EulerPhi[n]}, IntegerQ[(s + p)/2] && IntegerQ[Sqrt[s*p]] && IntegerQ[2 s*p/(s + p)]]; Select[Range[10^5], aQ]

A292063 Triangular numbers n such that psi(n) is also a triangular number, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 780, 2775, 5050, 474825, 681528, 1727011, 5286126, 5911641, 6604795, 17325441, 21612025, 27799696, 45025305, 386767578, 1538599128, 2086160121, 3679490220, 5718242211, 7092226351, 8019794628, 16505718895, 36604197735, 55541611986, 56693041356, 89369984476

Offset: 1

Amiram Eldar, Sep 08 2017

nonn

The indices of these triangular numbers are 1, 39, 74, 100, 974, 1167, 1858, 3251, 3438, 3634, 5886, 6574, 7456, 9489, ...

The indices of the triangular psi values are 1, 63, 95, 135, 1280, 1664, 2015, 4607, 4095, 4095, 7424, 7424, 9152, 12543, ...

			780 is in the sequence since 780 = 39*40/2 is triangular and psi(780) = 2016 = 63*64/2 is also triangular.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A000217, A001615, A083674, A083675, A115910, A116541, A287472.

psi[n_] := If[n<1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors @ n}]]; triQ[n_] := IntegerQ@ Sqrt[8n+1]; Select[Accumulate[Range[1000]], triQ[psi[#]]&]

a(18)-a(26) from Giovanni Resta, Sep 11 2017

A292064 Triangular numbers k such that psi(k) is a square, where psi(k) is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 66, 210, 276, 1128, 2346, 2556, 4278, 5778, 7140, 7750, 7875, 11781, 13041, 18336, 22578, 27966, 28920, 31878, 32131, 32640, 35511, 51681, 70125, 73536, 79800, 89676, 93096, 100128, 102378, 122760, 139128, 169653, 173755, 177906, 209628, 223446, 253116

Offset: 1

Amiram Eldar, Sep 08 2017

nonn

The indices of these triangular numbers are 1, 2, 11, 20, 23, 47, 68, 71, 92, 107, 119, 124, 125, 153, 161, 191, 212, 236, 240, ...
The indices of the square psi values are 1, 2, 12, 24, 24, 48, 72, 72, 96, 108, 144, 120, 120, 144, 144, 192, 216, 240, 264, ...
Intersection of A000217 and A291167. - Altug Alkan, Sep 08 2017

			66 is in the sequence since 66 = 11*12/2 is triangular, and psi(66) = 144 = 12^2 is square.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A001615, A287473, A291167.

psi[n_] := If[n<1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors @ n}]];
Select[Accumulate[Range[1000]], IntegerQ[Sqrt[psi[#]]]&]

cp's OEIS Frontend

A351455 a(n) = A064989(A001615(A003961(n))).

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A253628 Psi(n) mod n, where Psi is the Dedekind psi function (A001615).

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A291051 a(n) is the smallest number k such that psi(k) = n*phi(k) where psi(k) is Dedekind psi function (A001615) and phi(k) is Euler totient function (A000010), or 0 if no such k exists.

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A291138 a(n) is the smallest k such that psi(k) and phi(k) have same distinct prime factors when k is the product of n distinct primes (psi(k) = A001615(k) and phi(k) = A000010(k)), or 0 if no such k exists.

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A291205 Numbers n such that psi(n) = phi(m) has no solution for any m (psi(n) = A001615(n) and phi(m) = A000010(m)).

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A291209 Numbers n such that psi(n) is the sum of proper divisors of n where psi(n) = A001615(n).

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A291487 a(n) is the smallest k such that psi(k) = n!, or 0 if no such k exists (psi(k) = A001615(k)).

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A291959 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and psi(n) are all integers, where phi(n) is the Euler totient function (A000010) and psi(n) is the Dedekind psi function (A001615).

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